How far is the Sun from the center of the Milky Way?

It depends a bit on the particular 'distance indicator' you use. Here is a short summary of values given in a review article by astronomer Mark Reid at Harvard, published in the Annual Reviews of Astronomy and Astrophysics in 1993, vol. 31, p. 345: 


             Value in
Year        KiloParsecs          Method used
....................................................
1988          H2O Masers            7.1 +/- 1.5
1989        Globular Clusters       7.5 +/- 0.9
1982              "                 6.2 +/- 0.9
1980              "                10.1 +/- 0.7
1991        RR Lyrae Stars          8.2 +/- 1.0
1985              "                 8.0 +/- 0.7
1982        Giant Stars             7.9
1992          HII regions           7.7 +/- 0.9
1993          HII Regions           8.8 +/- 0.9
-----------------------------------------------------
The adopted, weighted average of all determinations is 8.0 +/- 0.5 kiloparsecs.
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